Optimal. Leaf size=24 \[ \frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )-\frac {1}{3} \tanh ^{-1}(x) \]
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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1144, 209, 213}
\begin {gather*} \frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )-\frac {1}{3} \tanh ^{-1}(x) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 213
Rule 1144
Rubi steps
\begin {align*} \int \frac {x^2}{-2+x^2+x^4} \, dx &=\frac {1}{3} \int \frac {1}{-1+x^2} \, dx+\frac {2}{3} \int \frac {1}{2+x^2} \, dx\\ &=\frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )-\frac {1}{3} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 32, normalized size = 1.33 \begin {gather*} \frac {1}{6} \left (2 \sqrt {2} \tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )+\log (1-x)-\log (1+x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.95, size = 25, normalized size = 1.04 \begin {gather*} -\frac {\text {Log}\left [1+x\right ]}{6}+\frac {\text {Log}\left [-1+x\right ]}{6}+\frac {\sqrt {2} \text {ArcTan}\left [\frac {\sqrt {2} x}{2}\right ]}{3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 26, normalized size = 1.08
method | result | size |
default | \(\frac {\arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}}{3}+\frac {\ln \left (-1+x \right )}{6}-\frac {\ln \left (1+x \right )}{6}\) | \(26\) |
risch | \(\frac {\arctan \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}}{3}+\frac {\ln \left (-1+x \right )}{6}-\frac {\ln \left (1+x \right )}{6}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.35, size = 25, normalized size = 1.04 \begin {gather*} \frac {1}{3} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {1}{6} \, \log \left (x + 1\right ) + \frac {1}{6} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 25, normalized size = 1.04 \begin {gather*} \frac {1}{3} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {1}{6} \, \log \left (x + 1\right ) + \frac {1}{6} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 29, normalized size = 1.21 \begin {gather*} \frac {\log {\left (x - 1 \right )}}{6} - \frac {\log {\left (x + 1 \right )}}{6} + \frac {\sqrt {2} \operatorname {atan}{\left (\frac {\sqrt {2} x}{2} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 38, normalized size = 1.58 \begin {gather*} \frac {\ln \left |x-1\right |}{6}-\frac {\ln \left |x+1\right |}{6}+\frac {4 \arctan \left (\frac {x}{\sqrt {2}}\right )}{2\cdot 3 \sqrt {2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.06, size = 17, normalized size = 0.71 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x}{2}\right )}{3}-\frac {\mathrm {atanh}\left (x\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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